Adaptive refinement and clustering of -matrices

Lars Grasedyck, Wolfgang Hackbusch, and Sabine Le Borne
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Submission date: 19. Dec. 2001
Pages: 14
MSC-Numbers: 65F05, 65F30, 65F50, 65N50
Keywords and phrases: hierarchical matrices, data-sparse approximations, formatted matrix operations, fast solvers, adaptive mesh refinement
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Abstract:
In previous papers, a class of (data-sparse) hierarchical matrices is introduced which allows an approximate matrix arithmetic of nearly optimal complexity. These so-called H-matrices were shown to be applicable in the boundary element as well as finite element context, again yielding nearly optimal complexity estimates for storage and work requirements of the respective stiffness matrices. The analyses were based on the assumption of the underlying cluster trees being balanced. This assumption might be violated in the case of adaptive mesh refinement. The present paper provides an extension of H-matrix techniques to (a sequence of) problems on adaptively refined meshes. A measure to monitor the actual storage and work complexities is introduced and employed to decide whether an adaptively refined (unbalanced) cluster tree is still acceptable or should be reconstructed in a balanced way.